
The first inequality is from triangle inequality of absolute value,
second inequality is trivial while the last inequality follows from
\(1-e^{-x}\le x\)  for any \(x\ge 0\) . To bound \(W_{2}\) , we use an
inequality from [1]}(Theorem 6.15, page 115):
\(W_{2}^{2}(\pi ,\ \pi _{\mu })\le 2\int _{\mathbb {R}^{d}}\Vert x\Vert _{2}^{2}\left|\pi (x)-\pi _{\mu }(x)\right|dx.\) 
